A fast tree algorithm for multi-component coagulation equation

TL;DR

This paper introduces a novel fast tree algorithm for multi-component coagulation equations, significantly reducing computational time and enabling detailed dust evolution modeling in protoplanetary disks.

Contribution

The authors developed a tree algorithm that reduces the complexity of multi-component coagulation calculations from exponential to near-linear, improving efficiency for dust evolution simulations.

Findings

The algorithm reduces computational complexity from $O(N^{2d})$ to $O(d N^d ext{log} N)$.

For $d=2$, the method is faster than traditional approaches across all tested parameters.

The algorithm maintains accuracy while significantly speeding up calculations, especially for higher dimensions.

Abstract

Dust properties, such as mass and porosity, impact planet formation directly. Understanding the time evolution of dust distribution across multiple properties requires numerical computation. However, available ways to calculate the multi-component coagulation-fragmentation are highly time-consuming. This study aims to develop a fast and accurate algorithm for multi-component coagulation. We assumed that two pairs of colliding aggregates reproduce a similar outcome if the dust properties are similar, and that the ratio of dust properties in logarithmic space gives the similarity as a "distance". These assumptions enable us to apply the tree algorithm, which groups distant bins and calculates interactions together, to coagulation. The algorithm reduces the computational complexity from $O (N^{2d})$ to $O (d N^d \log N)$, considering $N$ bins per $d$ components. We tested the algorithm by comparing it with the conventional direct method for cases where analytic solutions are known. We measured the dependencies of the wall-clock time, $L_2$ error in the distribution, and relative error of the total mass, on the $d, N$, opening angle $\theta_c$, and maximum dust distribution width after coagulation $k_c$. The algorithms are found to calculate coagulation consistently. For $d=1$, the tree method is faster than the direct method for a specific range of parameters. For $d=2$, however, the tree method is faster for all parameter regions surveyed, speeding it up by tens of times. Increasing $N$ and decreasing $\theta_c$ or $k_c$ made it slower and more accurate. Additionally, using a small $k_c$ performs worse than when using a large $k_c$, suggesting that limiting $k_c$ is unnecessary. We present a fast tree algorithm for the multi-component coagulation equation. It will enable us to evolve the multi-component dust distribution, such as in mass-porosity space, in protoplanetary disks.